{
  "id": 1191,
  "curve_key": "39980933217:-184593612800230929",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-832936109",
    "213650223049109"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    9
  ],
  "naive_height": 79.5138662323308,
  "faltings_height": 4.6839602219216845,
  "conductor": "14049882",
  "bad_primes": [
    "2",
    "3",
    "7",
    "11",
    "31",
    "109"
  ],
  "discriminant": "-19682229955776531123284168933376",
  "regulator": "14.4344489739583389452553415984744983031214",
  "points": [
    [
      "-3159",
      "14707018"
    ],
    [
      "-87283337/1369",
      "141101983096/50653"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Found by a rational-parameter search of the Kubert-Tate family for torsion 9, at t=-4/7. In y^2+(1-c)xy-by=x^3-bx^2 the parameters are defined by c=t^2*(t-1);b=c*(t^2-t+1); Converted to the global minimal model; PARI/GP ellrank returned bounds [2,2]. The 2 supplied points and the torsion subgroup are independently certified by the site.",
  "created_at": "2026-09-23 21:40:14",
  "updated_at": "2026-09-23 21:40:14"
}